On the fermionic T-duality of the AdS_4 \times CP^3 sigma-model
aa r X i v : . [ h e p - t h ] N ov Preprint typeset in JHEP style - HYPER VERSION
AEI-2010-128
On the fermionic T-duality of the
AdS × C P sigma-model Ido Adam
Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut)Am M¨uhlenberg 1, D-14476 Golm, Germany [email protected]
Amit Dekel and Yaron Oz
Raymond and Beverly Sackler School of Physics and AstronomyTel-Aviv University, Ramat-Aviv 69978, Israel [email protected] , [email protected] Abstract:
In this note we consider a fermionic T-duality of the coset realizationof the type IIA sigma-model on
AdS × C P with respect to the three flat directionsin AdS , six of the fermionic coordinates and three of the C P directions. We showthat the Buscher procedure fails as it leads to a singular transformation and discussthe result and its implications. Keywords:
Duality in Gauge Field Theories, String Duality. ontents
1. Introduction and summary 12. T-dualizing
AdS × C P
23. Discussion 5A. The osp(6 | superalgebra 6
1. Introduction and summary
Since the N = 6 superconformal Chern-Simons theory with matter was proposedby ABJM [1] as a dual to M-theory on AdS × S / Z k , which reduces in a certainlimit to the type IIA superstring on AdS × C P , much work has been devoted tounderstanding the properties of the ABJM field theory.Several tree-level scattering amplitudes of the ABJM theory were computed [2]and were shown to possess a Yangian symmetry, which includes the non-local chargesand the dual superconformal symmetry [3]. Some light-like polygonal Wilson loopsin the ABJM theory were computed in [4] and hinted that the ABJM theory mayhave a scattering amplitudes/Wilson loop duality, which would further support thecase in favor of the existence of dual superconformal symmetry. Additionally, acontour integral reproducing the known tree-level amplitudes has been recently pro-posed and was shown to have a Yangian symmetry [5]. Furthermore, a differentialrepresentation of a dual superconformal symmetry at tree-level has been constructed[6]. This representation involves variables dual to the ones parameterizing part ofthe R-symmetry in addition to the ones dual to the bosonic and fermionic momenta.The corresponding findings in N = 4 SYM in four dimensions were explainedfrom the point of view of string theory on AdS × S by a combination of bosonicand fermionic T-dualities, which is exact at the string tree-level [7, 8] (see [9] fora short review). Hence, it is interesting to see whether that is also the case fortype IIA strings on AdS × C P . Previously, it was found that the sigma-model for AdS × C P , realized as the coset OSp(6 | / (SO(2 , × U(3)) constructed in [10, 11],was not self-dual under T-duality involving both three directions in
AdS and sixfermionic coordinates [12, 13]. In fact, one could not perform a fermionic T-dualityin six fermionic isometries which together with the dualized bosonic ones form anAbelian subgroup of the whole isometry group.– 1 –n this note, in light of a suggestion that T-dualizing three isometries of C P is also required [3] and the new evidence [5, 6] from the field theory, we considerthe fermionic T-duality along the three flat AdS coordinates, three complex Killingvectors in C P (each one of real dimension one) as well as six of the fermionic co-ordinates, whose corresponding tangent-space vectors generate an Abelian subgroupof the isometry group. We show that as in the case of dualizing just in AdS and thefermions, the Buscher procedure fails as it leads to a singular transformation [12].The outline of this note is as follows: in Section 2 we apply the Buscher pro-cedure for T-duality to the OSp(6 | / (SO(2 , × U(3)) Green-Schwarz sigma-modeldescribing type IIA strings on
AdS × C P in a certain partial gauge-fixing and showthat it fails. In Section 3 we discuss the implications of the result. The osp(6 |
2. T-dualizing
AdS × C P We attempt to T-dualize
AdS × C P along the directions corresponding to P a , Q lα , R kl , which form an Abelian subalgebra of the isometry group.We assume that κ -symmetry can be partially gauge-fixed to set the six coordi-nates corresponding to ˆ S lα to zero and choose the coset representative g = e x a P a + θ lα Q lα + y kl R kl e B , e B = e ˆ θ αl ˆ Q lα + ξ lα S lα y D e ˆ y kl ˆ R kl , (2.1)where the indices a = 0 , , AdS , α = 1 , AdS spinor indices and l = 1 , , K = J + j , J = e − B ( dx a P a + dθ lα Q lα + dy kl R kl ) e B , j = e − B de B . (2.2)Examining the algebra, one finds that the current J takes values in the space spannedby { P a , Q lα , R kl , ˆ Q lα , λ kl , ˆ R kl } , while j is valued in span { ˆ Q lα , S lα , ˆ S lα , D, M ab , λ kl , ˆ R kl } .Denoting the decomposition of K into the Z -invariant subspaces by K i ∈ H i ,the Green-Schwarz action takes the form S = R πα ′ Z d z n − η ab J P a ¯ J P b − j D ¯ j D − J R kl ( ¯ J ˆ R kl + ¯ j ˆ R kl ) − J R kl ( J ˆ R kl + j ˆ R kl ) −− i C αβ h J Q lα ( ¯ J ˆ Q lβ + ¯ j ˆ Q lβ ) − ( J ˆ Q lα + j ˆ Q lα ) ¯ J Q lβ − j S lα ¯ j ˆ S lβ + j ˆ S lα ¯ j S lβ i o . (2.3)We attempt to T-dualize the action by using the Buscher procedure [14, 15] byintroducing the new fields A a , A lα , A kl , ¯ A a , ¯ A lα and ¯ A kl such that the current nowreads J = e − B ( A a P a + A lα Q lα + A kl R kl ) e B , (2.4)– 2 –hile j , which does not contain x a , θ lα and y kl , remains unmodified. In addition, thefollowing Lagrange multiplier terms are added to the action: S L = R πα ′ Z d z h ˜ x a ( ¯ ∂A a − ∂ ¯ A a ) + ˜ θ lα ( ¯ ∂A lα − ∂ ¯ A lα ) + ˜ y kl ( ¯ ∂A kl − ∂ ¯ A kl ) i , (2.5)where ˜ x a , ˜ θ lα and ˜ y kl are Lagrange multipliers.The T-duality is performed by integrating out the gauge fields, whose equationsof motion are0 = − η bc [ e − B P a e B ] P b J P c + i C αβ h [ e − B P a e B ] Q lα ( J ˆ Q lβ + j ˆ Q lβ ) −− [ e − B P a e B ] ˆ Q lα J Q lβ i − e − B P a e B ] R kl ( J ˆ R kl + j ˆ R kl ) − e − B P a e B ] ˆ R kl J R kl + ∂ ˜ x a , − η bc [ e − B Q lα e B ] P b J P c + i C βγ h [ e − B Q lα e B ] Q kβ ( J ˆ Q kγ + j ˆ Q kγ ) −− [ e − B Q lα e B ] ˆ Q kβ J Q kγ i − e − B Q lα e B ] R pq ( J ˆ R pq + j ˆ R pq ) − e − B Q lα e B ] ˆ R pq J R pq −− ∂ ˜ θ lα , − η bc [ e − B R kl e B ] P b J P c + i C αβ h [ e − B R kl e B ] Q pα ( J ˆ Q pβ + j ˆ Q pβ ) −− [ e − B R kl e B ] ˆ Q pα J Q pβ i − e − B R kl e B ] R pq ( J ˆ R pq + j ˆ R pq ) − e − B R kl e B ] ˆ R pq J R pq ++ ∂ ˜ y kl (2.6)for the holomorphic fields and0 = − η bc [ e − B P a e B ] P b ¯ J P c − i C αβ h [ e − B P a e B ] Q lα ( ¯ J ˆ Q lβ + ¯ j ˆ Q lβ ) − [ e − B P a e B ] ˆ Q lα ¯ J Q lβ i −− e − B P a e B ] R kl ( ¯ J ˆ R kl + ¯ j ˆ R kl ) − e − B P a e B ] ˆ R kl ¯ J R kl − ¯ ∂ ˜ x a , − η bc [ e − B Q lα e B ] P b ¯ J P c − i C βγ h [ e − B Q lα e B ] Q kβ ( ¯ J ˆ Q kγ + ¯ j ˆ Q kγ ) −− [ e − B Q lα e B ] ˆ Q kβ ¯ J Q kγ i − e − B Q lα e B ] R pq ( ¯ J ˆ R pq + ¯ j ˆ R pq ) − e − B Q lα e B ] ˆ R pq ¯ J R pq ++ ¯ ∂ ˜ θ lα , − η bc [ e − B R kl e B ] P b ¯ J P c − i C αβ h [ e − B R kl e B ] Q pα ( ¯ J ˆ Q pβ + ¯ j ˆ Q pβ ) −− [ e − B R kl e B ] ˆ Q pα ¯ J Q pβ i − e − B R kl e B ] R pq ( ¯ J ˆ R pq + ¯ j ˆ R pq ) − e − B R kl e B ] ˆ R pq ¯ J R pq −− ¯ ∂ ˜ y kl (2.7)for the anti-holomorphic ones. (The complexity of the equations arises from the factthat, unlike in the AdS × S case, J is valued in a space larger than the one that isactually dualized.)For the purpose of solving these equations, the properties of the field-dependentgroup-theoretic factors must be understood. In particular, it should be checkedwhether the coefficients of the gauge fields have non-trivial kernels.– 3 –n order to do so, we resort to explicitly expressing the currents in terms of thecoordinates. We denote C ≡ ˆ θ αl ˆ Q lα + ξ lα S lα and examine the commutators[ P a , C ] = − i √ γ aαβ ξ lα Q lβ ≡ Ξ P lβa Q lβ , [ Q lβ , C ] = 1 √ γ a C ) βα ˆ θ αl P a + 1 √ C βα ξ kα R lk ≡ Θ Qalβ P a + Ξ Qkβ R lk ≡ M lβ , [ R kl , C ] = − i √ θ αl δ pk − ˆ θ αk δ pl ) Q pα ≡ Θ Rpαkl Q pα . (2.8)We further define N lαkβ = Θ Qalα Ξ P kβa + Ξ
Qpα Θ Rkβpl (2.9)and note that [ M lα , C ] = N lαkβ Q kβ and [ Q lα , C ] = M lα . Using the formula e − B Ae B = A + [ A, B ] + [[ A, B ] , B ] + . . . , we get e − C ( dx a P a + dθ lα Q lα + dy kl R kl ) e C = dx a P a + dy kl R kl ++ ( dx a Ξ P lαa + dy pq Θ Rlαpq ) cosh √ N − N ! lαkβ M kβ + sinh √ N √ N ! lαkβ Q kβ ++ dθ lα sinh √ N √ N ! lαkβ M kβ + (cosh √ N ) lαkβ Q kβ . (2.10)Finally, conjugating with y D e ˆ y kl ˆ R kl yields the current J = dx a y P a + dy kl ( R kl + 2 i √ y kq λ lq + 2ˆ y kq ˆ y ln ˆ R qn )++ ( dx a Ξ P lαa + dy pq Θ Rlαpq ) cosh √ N − N ! lαkβ + dθ lα sinh √ N √ N ! lαkβ ×× h ˜ M kβ + i √ Qmβ (ˆ y kq λ mq − ˆ y mq λ kq ) + Ξ Qrβ (ˆ y kq ˆ y rn − ˆ y rq ˆ y kn ) ˆ R qn i ++ 1 y / ( dx a Ξ P lαa + dy pq Θ Rlαpq ) sinh √ N √ N ! lαkβ + dθ lα (cosh √ N ) lαkβ ×× ( Q kβ + i √ y pk ˆ Q pβ ) , (2.11)where ˜ M kβ ≡ y − D M kβ y D = y Θ Qalα P a + Ξ Qlα R kl .Unfortunately, j is even more complicated. However, before plunging into itscomputation in a closed form it is worthwhile to examine it to the lowest order in ˆ θ αl and ξ lα . Doing so yields, j = d ˆ θ αl y / ˆ Q lα + y / dξ lα S lα − i √ y / ˆ y kl dξ lα ˆ S kα + dyy D + d ˆ y pq ˆ R pq + O (ˆ θ αl , ξ lα ) . (2.12)– 4 –aving the currents, we can take a look at the action to lowest order in ˆ θ αl and ξ lα : S = R πα ′ Z d z n − η ab ∂x a ¯ ∂x b y − ∂y ¯ ∂yy − ∂y kl (2ˆ y pk ˆ y ql ¯ ∂y pq + ¯ ∂ ˆ y kl ) −− ∂y kl (2ˆ y pk ˆ y ql ∂y pq + ∂ ˆ y kl ) − i y C αβ h ∂θ lα ( i √ y kl ¯ ∂θ kβ + ¯ ∂ ˆ θ βl ) −− ( i √ y kl ∂θ kα + ∂ ˆ θ αl ) ¯ ∂θ lβ i + i yC αβ ( − i √ y lk ∂ξ lα ¯ ∂ξ kβ + i √ y lk ∂ξ kα ¯ ∂ξ lβ ) o . (2.13)The term quadratic in the θ lα derivatives is multiplied by a three-dimensional an-tisymmetric matrix, whose rank is two, and the higher order terms in ˆ θ αl and ξ lα cannot make the matrix’s kernel trivial. Thus the term quadratic in the fermionicgauge fields in the dualized action will be multiplied by a singular matrix and thefermionic gauge fields will be multiplied by a singular matrix in the equations ofmotion — one cannot T-dualize all the six fermionic coordinates.Since the obstruction to T-dualizing the fermionic coordinates is at the zerothorder in the spectator fermions, it appears that modifying the κ -symmetry gauge-fixing of these fermionic degrees of freedom would not change the above conclusion.
3. Discussion
We showed that the application of the Buscher T-duality procedure to the cosetOSp(6 | / (SO(2 , × U(3)) fails when dualizing along the
AdS flat directions, threeof the (real) C P directions and six fermionic directions. There are several ways toexplain this apparent tension between the field theory tree-level evidence and thesigma-model analysis.The simplest and most obvious explanation is that the dual superconformalsymmetry exists only in the weakly-coupled field theory description and breaks downat the strong-coupling regime, which is described by the string theory dual. A secondpossibility is that in this case the dual superconformal symmetry is not related tothe ordinary superconformal symmetry by a T-duality transformation but in a moreintricate way.A third possibility is that the coset formulation does not capture the entire super-string description. The coset is obtained by a partial gauge-fixing of the κ -symmetryof the full AdS × C P sigma-model [16] by setting the fermionic coordinates corre-sponding to the eight broken supersymmetries to zero. However, as noted in [16], thisgauge-fixing is not compatible with all the possible string configurations. Thus, itdoes not have a representation for certain field theory operators, which might amountto a (possibly inconsistent) truncation of the field theory that does not preserve thedual superconformal symmetry. A way to resolve this issue could be to use a bettergauge-fixing of the κ -symmetry as proposed in [16, 13].– 5 – cknowledgments We would like to thank Y-t. Huang and A. E. Lipstein for sharing a draft of theirpaper [6] with us before its publication. I.A. is supported in part by the German-Israeli Project cooperation (DIP H.52) and the German-Israeli Fund (GIF).
A. The osp(6 | superalgebra The osp(6 |
4) algebra’s commutation relations in the so(1 , ⊕ u(3) basis are given by[ λ kl , λ mn ] = i √ δ ml λ kn − δ kn λ ml ) , (A.1)[ λ kl , R mn ] = i √ δ ml R kn − δ nl R km ) , [ λ lk , ˆ R pq ] = − i √ δ pl ˆ R kq − δ ql ˆ R kp ) (A.2)[ R mn , R kl ] = 0 , [ R mn , ˆ R kl ] = i √ δ mk λ nl − δ ml λ nk − δ nk λ ml + δ nl λ mk ) (A.3)[ P a , P b ] = 0 , [ K a , K b ] = 0 , [ P a , K b ] = η ab D − M ab (A.4)[ M ab , M cd ] = η ac M bd + η bd M ac − η ad M bc − η bc M ad (A.5)[ M ab , P c ] = η ac P b − η bc P a , [ M ab , K c ] = η ac K b − η bc K a (A.6)[ D, P a ] = P a , [ D, K a ] = − K a , [ D, M ab ] = 0 (A.7)[ D, Q lα ] = 12 Q lα , [ D, S lα ] = − S lα (A.8)[ P a , Q lα ] = 0 , [ K a , S lα ] = 0 (A.9)[ P a , S lα ] = − i √ γ a ) αβ Q lβ , [ K a , Q lα ] = i √ γ a ) αβ S lβ (A.10)[ M ab , Q lα ] = − i γ ab ) αβ Q lβ , [ M ab , S lα ] = − i γ ab ) αβ S lβ (A.11)[ R kl , ˆ Q pα ] = i √ δ lp Q kα − δ kp Q lα ) , [ R kl , ˆ S pα ] = − i √ δ lp S kα − δ kp S lα ) (A.12)[ ˆ R kl , Q pα ] = − i √ δ pl ˆ Q kα − δ pk ˆ Q lα ) , [ ˆ R kl , S pα ] = i √ δ pl ˆ S kα − δ pk ˆ S lα ) (A.13)[ λ kl , Q pα ] = i √ δ pl Q kα , [ λ kl , S pα ] = i √ δ pl S kα (A.14)[ λ kl , ˆ Q pα ] = − i √ δ kp ˆ Q lα , [ λ kl , ˆ S pα ] = − i √ δ kp ˆ S lα (A.15) { Q lα , Q kβ } = 0 , { Q lα , ˆ Q kβ } = − √ δ lk ( γ a C ) αβ P a (A.16) { S lα , S kβ } = 0 , { S lα , ˆ S kβ } = − √ δ lk ( γ a C ) αβ K a (A.17)– 6 – Q lα , S kβ } = − √ C αβ R lk , { ˆ Q lα , ˆ S kβ } = − √ C αβ ˆ R lk (A.18) { Q lα , ˆ S kβ } = − i δ lk ( C αβ D + i
12 ( γ ab C ) αβ M ab ) + 1 √ C αβ λ lk (A.19) { ˆ Q lα , S kβ } = i δ kl ( C αβ D − i
12 ( γ ab C ) αβ M ab ) + 1 √ C αβ λ kl (A.20)The indices take the values k, l = 1 , ...,
3, the u(3), a, b = 0 , , of so(1 , α, β, .. = 1 , ,
1) spinors, and η = diag( − , + , +). The generatorssatisfy the following relations under complex conjugation R ∗ kl = ˆ R kl , λ kl = λ ∗ l k ,ˆ Q lα = ( Q lα ) ∗ and ˆ S lα = ( S lα ) ∗ . The ( γ a ) αβ are the Dirac matrices of so(1 , γ ab = i [ γ a , γ b ]. We raise and lower spinor indices using C αβ = ǫ αβ , ψ α = ψ β ǫ βα , ψ α = ǫ αβ ψ β , where ǫ = − ǫ = ǫ = − ǫ = 1.The bilinear forms are given byStr( R kl , ˆ R pq ) = δ kq δ lp − δ kp δ lq , Str( λ kl , λ pq ) = − δ kq δ lp , Str( Q lα , ˆ S kβ ) = iδ lk C αβ , Str( S lα , ˆ Q kβ ) = − iδ kl C αβ , Str( P a , K b ) = − η ab , Str(
D, D ) = − , Str( M ab , M cd ) = η ac η bd − η ad η bc . (A.21)The Z subspaces with the invariant locus of U(3) × SO(3 ,
1) which gives thesemi-symmetric space
AdS × C P are H = { P a − K a , M ab , λ kl } , H = { Q lα − S lα , ˆ Q lα − ˆ S lα } , H = { P a + K a , D, R kl , ˆ R kl } , H = { Q lα + S lα , ˆ Q lα + ˆ S lα } . (A.22) References [1] O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, “N=6 superconformalChern-Simons-matter theories, M2-branes and their gravity duals,”
JHEP (2008)091, arXiv:0806.1218 [hep-th] .[2] A. Agarwal, N. Beisert, and T. McLoughlin, “Scattering in Mass-Deformed N ≥ JHEP (2009) 045, arXiv:0812.3367 [hep-th] .[3] T. Bargheer, F. Loebbert, and C. Meneghelli, “Symmetries of Tree-level ScatteringAmplitudes in N=6 Superconformal Chern-Simons Theory,” arXiv:1003.6120[hep-th] .[4] J. M. Henn, J. Plefka, and K. Wiegandt, “Light-like polygonal Wilson loops in 3dChern-Simons and ABJM theory,” arXiv:1004.0226 [hep-th] . – 7 –
5] S. Lee, “Yangian Invariant Scattering Amplitudes in Super-Chern- Simons Theory,” arXiv:1007.4772 [hep-th] .[6] Y.-t. Huang and A. E. Lipstein, “Dual Superconformal Symmetry of N=6Chern-Simons Theory,” arXiv:1008.0041 [hep-th] .[7] N. Berkovits and J. Maldacena, “Fermionic T-Duality, Dual SuperconformalSymmetry, and the Amplitude/Wilson Loop Connection,”
JHEP (2008) 062, arXiv:0807.3196 [hep-th] .[8] N. Beisert, R. Ricci, A. A. Tseytlin, and M. Wolf, “Dual Superconformal Symmetryfrom AdS5 x S5 Superstring Integrability,” arXiv:0807.3228 [hep-th] .[9] N. Beisert, “T-Duality, Dual Conformal Symmetry and Integrability for Strings on AdS × S ,” Fortschr. Phys. (2009) 329–337, arXiv:0903.0609 [hep-th] .[10] j. Stefanski, B., “Green-Schwarz action for Type IIA strings on AdS × CP ,” Nucl.Phys.
B808 (2009) 80–87, arXiv:0806.4948 [hep-th] .[11] G. Arutyunov and S. Frolov, “Superstrings on
AdS xCP as a Coset Sigma-model,” JHEP (2008) 129, arXiv:0806.4940 [hep-th] .[12] I. Adam, A. Dekel, and Y. Oz, “On Integrable Backgrounds Self-dual underFermionic T- duality,” JHEP (2009) 120, arXiv:0902.3805 [hep-th] .[13] P. A. Grassi, D. Sorokin, and L. Wulff, “Simplifying superstring and D-brane actionsin AdS(4) x CP(3) superbackground,” JHEP (2009) 060, arXiv:0903.5407[hep-th] .[14] T. H. Buscher, “A Symmetry of the String Background Field Equations,” Phys. Lett.
B194 (1987) 59.[15] T. H. Buscher, “Path Integral Derivation of Quantum Duality in Nonlinear SigmaModels,”
Phys. Lett.
B201 (1988) 466.[16] J. Gomis, D. Sorokin, and L. Wulff, “The complete AdS(4) x CP(3) superspace forthe type IIA superstring and D-branes,” arXiv:0811.1566 [hep-th] ..